$\backslash$table$[[$PART $1,],[,,$Cross$-$sectional areas, $A(m{m}^2)$

Task:

Part $1$: Beam analysis

The timber beam in Figure $1$ is simply supported at points A and E $.$ The cross$-$section of the beam changes from segment AC to CE$,$ as shown below. The Young's modulus of timber may be taken as $12$ GPa. The load, geometry and cross$-$sections of the beam will be instructed to each student in class.

Figure $1$

$($a$)$ Determine the deflection of the beam at point D by the Virtual Work Method. $[20$ Marks$]$

$($b$)$ Without showing any calculations, briefly explain the difference in approach if Castigliano's Theorem were used in part $($a$),$ instead of the Virtual Work Method. $[5$ Marks$]$

$($c$)$ Calculate the bending and shear stresses at point F on the Section B$-$B$.[15$ Marks$]$

$($d$)$ Sketch the Mohr's circle of stresses corresponding to point F$.[10$ Marks$]$